SHEAVES OF MICROFUNCTIONS
Andrea D’Agnolo Giuseppe Zampieri
Abstract. Let A be a closed set of M ∼ = R
n, whose conormal cones x + γ
x∗(A), x ∈ A, have locally empty intersection. We first show in §1 that dist(x, A), x ∈ M \ A is a C
1function. We then represent the microfunctions of C
A|X, X ∼ = C
n, (cf [S]), using cohomology groups of O
Xof degree 1. By the results of §1–3, we are able to prove in §4 that the sections of C
A|X˛
˛
(TM∗X⊕γ∗(A))x0, x
0∈ ∂A, satisfy the princi- ple of analytic continuation in the complex integral manifolds of {H(φ
Ci)}
i=1,...,m, {φ
i} being a base for the linear hull of γ
∗x0(A) in T
x∗0M; in particular we get
Γ
A×MT∗MX
(C
A|X) ˛
˛
˛
∂A×MT˙M∗X= 0. In the proof we deeply use a variant of Bochner’s theorem due to [Kan]. When A is a half space with C
ωboundary, all above results were already proved by Kataoka in [Kat 1]. Finally for a E
X-module M we show that Hom
EX(M, C
A|X)
p= 0, p ∈ ˙π
−1(x
0), when at least one conormal θ ∈ ˙γ
x∗0(A) is non-characteristic for M. We also show that for an open domain Ω such that the set A = M \ Ω verifies the above conditions, we have (C
Ω|X)
T∗MX